I am trying to solve the Non-linear Schrodinger equation

$-\Delta \psi(r) + \psi(r) - |\psi(r)|^2\psi(r) = 0$ where $r \in \Omega$

In a square domain ($(x,y) \in \Omega$ where $\Omega=[0,1]\times [0,1]$) with the Dirichlet condition

$\psi(r) = 0$ for $r \in \partial \Omega$

and with the additional constraint

$\int_\Omega d^2r \, |\psi(r)|^2 = 1$

Does somebody know how to include the integral condition into NDSolve?

Thanks in advance


  • 1
    $\begingroup$ Hola Alvaro, welcome to Mathematica.SE, please consider taking the tour so you learn the basic rules of the site. Here its considered helpful to share your code attempts, in a well formatted easy to copy&paste form. Don't erase your nicely formatted $\LaTeX$, just add Mathematica code in addition. $\endgroup$ – rhermans Oct 12 '14 at 16:20
  • $\begingroup$ Whoever wrote this page en.wikipedia.org/wiki/Nonlinear_Schrödinger_equation used mathematica... $\endgroup$ – chris Oct 12 '14 at 17:05
  • $\begingroup$ You might also want to look at demonstrations.wolfram.com/… though this does not look very promising stackoverflow.com/questions/25502814/inverse-scattering-method $\endgroup$ – chris Oct 12 '14 at 17:09
  • $\begingroup$ A Lagrange multiplier doesn't work? $\endgroup$ – yohbs Oct 13 '14 at 7:28
  • $\begingroup$ Hello, actually my first attempt was using a Lagrange multiplier, but NDSolve accepts only algebraic constraints, as for example: $f(r)+g(r)=1$. To put it differently it seems to me that it only accepts local constraints and not non-local as $\int_\Omega d^2 r \, |\psi(r)|^2 = 1$. In fact my question is precisely how to circumvent this problem. Thanks $\endgroup$ – Alvaro Rojo Bravo Oct 13 '14 at 13:54

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